metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.180D6, C6.412- (1+4), C6.862+ (1+4), C4⋊Q8⋊18S3, C4⋊C4.127D6, D6⋊Q8⋊49C2, D6⋊3Q8⋊39C2, (C2×Q8).114D6, C42⋊3S3⋊20C2, Dic3.Q8⋊43C2, (C2×C6).279C24, D6⋊C4.76C22, D6.D4.5C2, Dic3⋊Q8⋊28C2, C2.90(D4⋊6D6), (C2×C12).641C23, (C4×C12).274C22, C12.23D4.9C2, (C6×Q8).146C22, (C2×D12).174C22, Dic3⋊C4.88C22, C4⋊Dic3.256C22, C22.300(S3×C23), (C22×S3).124C23, C2.42(Q8.15D6), C3⋊6(C22.57C24), (C2×Dic3).147C23, (C4×Dic3).168C22, (C2×Dic6).193C22, (C3×C4⋊Q8)⋊21C2, C4⋊C4⋊S3⋊48C2, (S3×C2×C4).152C22, (C3×C4⋊C4).222C22, (C2×C4).222(C22×S3), SmallGroup(192,1294)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 480 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×13], C22, C22 [×6], S3 [×2], C6, C6 [×2], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4, Q8 [×3], C23 [×2], Dic3 [×6], C12 [×7], D6 [×6], C2×C6, C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8 [×2], C2×Q8, Dic6, C4×S3 [×2], D12, C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×2], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8, C4⋊Q8, C4×Dic3 [×2], Dic3⋊C4 [×10], C4⋊Dic3 [×2], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C6×Q8 [×2], C22.57C24, C42⋊3S3 [×2], Dic3.Q8 [×2], D6.D4 [×2], D6⋊Q8 [×2], C4⋊C4⋊S3 [×2], Dic3⋊Q8, D6⋊3Q8 [×2], C12.23D4, C3×C4⋊Q8, C42.180D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4), 2- (1+4) [×2], S3×C23, C22.57C24, D4⋊6D6, Q8.15D6 [×2], C42.180D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 42 7 48)(2 37 8 43)(3 44 9 38)(4 39 10 45)(5 46 11 40)(6 41 12 47)(13 29 19 35)(14 36 20 30)(15 31 21 25)(16 26 22 32)(17 33 23 27)(18 28 24 34)(49 69 55 63)(50 64 56 70)(51 71 57 65)(52 66 58 72)(53 61 59 67)(54 68 60 62)(73 92 79 86)(74 87 80 93)(75 94 81 88)(76 89 82 95)(77 96 83 90)(78 91 84 85)
(1 25 55 84)(2 73 56 26)(3 27 57 74)(4 75 58 28)(5 29 59 76)(6 77 60 30)(7 31 49 78)(8 79 50 32)(9 33 51 80)(10 81 52 34)(11 35 53 82)(12 83 54 36)(13 61 95 40)(14 41 96 62)(15 63 85 42)(16 43 86 64)(17 65 87 44)(18 45 88 66)(19 67 89 46)(20 47 90 68)(21 69 91 48)(22 37 92 70)(23 71 93 38)(24 39 94 72)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 83 31 77)(26 76 32 82)(27 81 33 75)(28 74 34 80)(29 79 35 73)(30 84 36 78)(37 67 43 61)(38 72 44 66)(39 65 45 71)(40 70 46 64)(41 63 47 69)(42 68 48 62)(49 54 55 60)(50 59 56 53)(51 52 57 58)(85 96 91 90)(86 89 92 95)(87 94 93 88)
G:=sub<Sym(96)| (1,42,7,48)(2,37,8,43)(3,44,9,38)(4,39,10,45)(5,46,11,40)(6,41,12,47)(13,29,19,35)(14,36,20,30)(15,31,21,25)(16,26,22,32)(17,33,23,27)(18,28,24,34)(49,69,55,63)(50,64,56,70)(51,71,57,65)(52,66,58,72)(53,61,59,67)(54,68,60,62)(73,92,79,86)(74,87,80,93)(75,94,81,88)(76,89,82,95)(77,96,83,90)(78,91,84,85), (1,25,55,84)(2,73,56,26)(3,27,57,74)(4,75,58,28)(5,29,59,76)(6,77,60,30)(7,31,49,78)(8,79,50,32)(9,33,51,80)(10,81,52,34)(11,35,53,82)(12,83,54,36)(13,61,95,40)(14,41,96,62)(15,63,85,42)(16,43,86,64)(17,65,87,44)(18,45,88,66)(19,67,89,46)(20,47,90,68)(21,69,91,48)(22,37,92,70)(23,71,93,38)(24,39,94,72), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,83,31,77)(26,76,32,82)(27,81,33,75)(28,74,34,80)(29,79,35,73)(30,84,36,78)(37,67,43,61)(38,72,44,66)(39,65,45,71)(40,70,46,64)(41,63,47,69)(42,68,48,62)(49,54,55,60)(50,59,56,53)(51,52,57,58)(85,96,91,90)(86,89,92,95)(87,94,93,88)>;
G:=Group( (1,42,7,48)(2,37,8,43)(3,44,9,38)(4,39,10,45)(5,46,11,40)(6,41,12,47)(13,29,19,35)(14,36,20,30)(15,31,21,25)(16,26,22,32)(17,33,23,27)(18,28,24,34)(49,69,55,63)(50,64,56,70)(51,71,57,65)(52,66,58,72)(53,61,59,67)(54,68,60,62)(73,92,79,86)(74,87,80,93)(75,94,81,88)(76,89,82,95)(77,96,83,90)(78,91,84,85), (1,25,55,84)(2,73,56,26)(3,27,57,74)(4,75,58,28)(5,29,59,76)(6,77,60,30)(7,31,49,78)(8,79,50,32)(9,33,51,80)(10,81,52,34)(11,35,53,82)(12,83,54,36)(13,61,95,40)(14,41,96,62)(15,63,85,42)(16,43,86,64)(17,65,87,44)(18,45,88,66)(19,67,89,46)(20,47,90,68)(21,69,91,48)(22,37,92,70)(23,71,93,38)(24,39,94,72), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,83,31,77)(26,76,32,82)(27,81,33,75)(28,74,34,80)(29,79,35,73)(30,84,36,78)(37,67,43,61)(38,72,44,66)(39,65,45,71)(40,70,46,64)(41,63,47,69)(42,68,48,62)(49,54,55,60)(50,59,56,53)(51,52,57,58)(85,96,91,90)(86,89,92,95)(87,94,93,88) );
G=PermutationGroup([(1,42,7,48),(2,37,8,43),(3,44,9,38),(4,39,10,45),(5,46,11,40),(6,41,12,47),(13,29,19,35),(14,36,20,30),(15,31,21,25),(16,26,22,32),(17,33,23,27),(18,28,24,34),(49,69,55,63),(50,64,56,70),(51,71,57,65),(52,66,58,72),(53,61,59,67),(54,68,60,62),(73,92,79,86),(74,87,80,93),(75,94,81,88),(76,89,82,95),(77,96,83,90),(78,91,84,85)], [(1,25,55,84),(2,73,56,26),(3,27,57,74),(4,75,58,28),(5,29,59,76),(6,77,60,30),(7,31,49,78),(8,79,50,32),(9,33,51,80),(10,81,52,34),(11,35,53,82),(12,83,54,36),(13,61,95,40),(14,41,96,62),(15,63,85,42),(16,43,86,64),(17,65,87,44),(18,45,88,66),(19,67,89,46),(20,47,90,68),(21,69,91,48),(22,37,92,70),(23,71,93,38),(24,39,94,72)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,83,31,77),(26,76,32,82),(27,81,33,75),(28,74,34,80),(29,79,35,73),(30,84,36,78),(37,67,43,61),(38,72,44,66),(39,65,45,71),(40,70,46,64),(41,63,47,69),(42,68,48,62),(49,54,55,60),(50,59,56,53),(51,52,57,58),(85,96,91,90),(86,89,92,95),(87,94,93,88)])
Matrix representation ►G ⊆ GL10(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(10,GF(13))| [12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,8,0,5,0,0,0,0,0,0,3,1,1,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0],[0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,5,0,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0],[12,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,5,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4G | 4H | ··· | 4M | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | 2+ (1+4) | 2- (1+4) | D4⋊6D6 | Q8.15D6 |
| kernel | C42.180D6 | C42⋊3S3 | Dic3.Q8 | D6.D4 | D6⋊Q8 | C4⋊C4⋊S3 | Dic3⋊Q8 | D6⋊3Q8 | C12.23D4 | C3×C4⋊Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{180}D_6 % in TeX
G:=Group("C4^2.180D6"); // GroupNames label
G:=SmallGroup(192,1294);
// by ID
G=gap.SmallGroup(192,1294);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,100,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations